Analysis of Eclipsing Binary Data

analysis of eclipsing binary data

A.E. Lynas-Gray and C. Causer October 2012 AS33

1 Introduction

The purpose of this experiment is to verify physical and geometric characteristics of an eclipsing binary already published in the literature, using radial velocity and light curves provided. In carrying out the experiment, the most important thing is to appreciate how the masses, radii, shapes, temperatures and orbits of the two stars determine the observed light and radial velocity variations. The book by Hilditch (2001) is an excellent introduction to binary stars and is recommended reading.

Once the period has been determined, masses, radii and effective temperatures of both components are estimated using a “back of an envelope” calculation; these results are then used as a starting approximation in a light and radial velocity curve synthesis code. The idea is to reproduce observed light and radial velocity curves, but in practice “back of the envelope” results will need adjustment before a good agreement with observation is achieved. At the time of writing, only data for R Canis Majoris are provided.

2 Initial setup

Begin by creating a xterm window in which commands can be typed. The window is created using the Mouse to click on the large “X” in the middle of the dock on the bottom of your screen. The large “X” may bob up and down a bit before the window appears.

The steps detailed below provide the setup necessary for experiments carried out in the Astrophysics Laboratory. The .profile, .bashrc, .cups/lpoptions and .Xdefaults files in your home directory are changed as a result of carrying out steps listed below; if these files have been previously setup to carry out an experiment in another laboratory, please consult a demonstrator before proceeding. If you have already carried an experiment in the Astrophysics Laboratory and the .profile, .bashrc, .cups/lpoptions and .Xdefaults files in your home directory have remained unchanged since, skip the steps below and move on to the next section.

Provide yourself with copies of the necessary .profile, .bashrc, .cups/lpoptions and .Xdefaults ﬁles by executing the commands

$ cp ~aelg/astro-lab/profile .profile $ cp ~aelg/astro-lab/bashrc .bashrc

03Oct2012 Copyright c 2012 University of Oxford, except where indicated. AS33- 1 Analysis of eclipsing binary data

$ cp ~aelg/astro-lab/Xdefaults .Xdefaults $ cp ~aelg/astro-lab/lpoptions .cups/lpoptions

where the leading ”$” represents the operating system command line prompt which by default is some long string. Once the above commands have been typed, logout and login again; this causes the .profile, .bashrc, .cups/lpoptions and .Xdefaults ﬁles to be executed and so providing the setup necessary to conduct experiments in the Astrophysics Laboratory. Create a new xterm window, as described above, in which commands can be typed; the ﬁrst obvious sign of the new setup is the replacement of the default operating system command line prompt with a simple “$”.

It is worth noting some unorthodox inputting methods inherent to Mac keyboards and mice. Some computers in the Lab have mice with a right click feature (which is turned on in “System Preferences.”), For mice without this feature (the clear ones), a right click can be emulated by using ctrl + left click. Also, programs such as NIGHTFALL use the # symbol in their data ﬁles; this is typed by pressing shift + 3 (the £ symbol is typed by pressing alt + 3).

3 Getting started

All computer commands reproduced above and below for your guidance are given in typewriter font. The leading "$ " represents the command line prompt. A leading “> ” represents a prompt provided by an executing program such as dipso. Typewriter font is also used to specify ﬁle names, ﬁle contents and environment variables. Further necessary information is provided in the Third Year Astrophysics Laboratory Practicals page

http://www-astro.physics.ox.ac.uk/~aelg/Third_Year_Laboratory

which can be read with a browser such as safari.

The ﬁrst step is to create a new directory in which all ﬁles, associated with the experiment described here, will be stored. Issue a command of the form

$ mkdir as33

to create such a directory. Then type

$ cd as33

to make the new directory your current working directory.

Data files will need to be edited as described below, or a decision (justified in the material submitted for marking) may be taken to discard one or more observations. Write access to the data files is also required by the programs used although the files are not changed by them. Copies of the light curve and radial velocity files are therefore made in the working directory. The commands issued depend on the star being analysed; in the case of R Canis Majoris

$ cp ~aelg/eclipsing_binary/data/R_CMA/B.dat .

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$ cp ~aelg/eclipsing_binary/data/R_CMA/V.dat . $ cp ~aelg/eclipsing_binary/data/R_CMA/rv1.dat . $ cp ~aelg/eclipsing_binary/data/R_CMA/rv2.dat .

are typed; the full stop at the end specifies that the file is to have the same name when copied to the current directory. Each file consists of three columns which from left to right give time (as Heliocentric Julian date expressed in days), the observation (in magnitudes or kilometres per second) and the standard deviation in the observation (in the same units as used for the observation). The files B.dat and V.dat contain respectively Johnson B-band and V-band differential photometry; rv1.dat and rv2.dat contain radial velocities for the more massive and less massive star respectively.

In the case of R Canis Majoris, differential photometry is taken from Radhakrishnan & Sarma (1982) and radial velocities from Tomkin (1985). Radhakrishnan & Sarma use BD −15o1732 as their comparison star and for which they obtain V = 5.48 ± 0.01, (B − V) = 0.07 ± 0.01 and (U − B) = 0.04 ± 0.01. The epoch (see below) adopted by Radhakrishnan & Sarma, and to be used here for phasing all observations, is 2444648.3283. Ribas et al. (2002) give a good overview R Canis Majoris, along with references to earlier work,

4 Period determination

Eclipsing binaries generally have light curves which exhibit well deﬁned Primary Minima and the time intervals between these events can be a good indicator of the period; it is never an exact indicator, of course, because it is unlikely that an observation was ever made at the exact time of Primary Minimum. Write down the approximate times of Primary Minima as seen in the Johnson B-band and Johnson V-band light curves and look at the intervals between them. The smallest interval between any two Primary Minima is either the period or some integral multiple of it.

Try to improve the estimate of the period by taking the average of the smallest intervals. Are the larger intervals exact multiples of the smaller intervals? If not, use a larger interval to obtain a more precise estimate of the period. How can the possibility that the derived period is a multiple of the true period be excluded?

Once a period has been determined, phases for each observation can be calculated using a relation of the form φ = (T − T0)/P (1)

where P is the period in days, T is the Heliocentric Julian date of observation (also in days) and T0 is the Heliocentric Julian date of some reference observation (also in days) at which the phase is deﬁned to be zero. In the case of an eclipsing binary, it makes sense to select T0 as a time of Primary Minimum so that when the correct period is used all Primary Minima are at φ = 0 or integral values of φ.

Because the times of Primary Minima are not measured precisely, the period derived using them can at best be a starting approximation for an improved technique which uses all observations. The improved technique adopted here is the “string-length” method which Dworetsky (1983) proposes. Once the correct period is used in (1), the plot of observation (magnitude or radial velocity) as a function of phase results in a smooth curve; that is the the length of the piecewise line joining each point (or “string- length”) is minimised.

The program strphase plots observations as a function of phase for a range of trial periods; in each case it also gives the “string-length” (l) calculated using

n−1 1/2 1/2 h 2 2i h 2 2i l = ∑ (oi − oi−1) + (φi − φi−1) + (o1 − on) + (1 + φ1 − φn) (2) i=2

th where oi is the i observation scaled as Dworetsky (1983) recommends. In calculating the standard deviation (σ(l)) in the “string-length” using strphase, it is assumed that each trial phase is known exactly. Moreover, each observation is independent and so uncorrelated with any other; we therefore have n 2 2 ∂l 2 σ (l) = 2 ∑ σ (oi) (3) i=1 ∂oi th 2 th where the variance in the i scaled observation (σ (oi)) is related to the variance in the unscaled i 2 observation (σ (Oi)) by

2 2 2 2 ∂oi 2 ∂oi 2 ∂oi 2 σ (oi) = σ (Oi) + σ (Omax) + σ (Omin) (4) ∂Oi ∂Omax ∂Omin with ∂o 1 i = , (5) ∂Oi 2(Omax − Omin) ∂o O − O i = i max 2 , (6) ∂Omin 2(Omax − Omin) and ∂o O − O i = − i min 2 . (7) ∂Omax 2(Omax − Omin)

Here Omax and Omin are respectively the maximum and minimum unscaled observations.

Before running strphase it is necessary to tell it which set of observations to use; this is done through setting an environment variable. For example, the command

$ OBSERVATIONS=B.dat; export OBSERVATIONS would be used if data in the ﬁle B.dat were to be used in the period determination. The program strphase is then started by simply typing

$ strphase and supplying responses to the questions asked. The default epoch may be selected during the period determination exercise, but the recommended epoch must be used for the preparation of light and radial velocity curves to be submitted for marking.

It is suggested that a wide period range be selected initially, with a period increment chosen so that something like ten or twenty periods will be tried in each run. At the end of a run a summary file period-determination.dat nnnnn is written in the working directory, where nnnnn is the run number starting at zero. The quality of each trial period can be assessed from plots made during a run but the summary file is also useful; in particular, look for a period which gives a “string-length” significantly (several standard deviations) smaller than any other. A smaller range of trial periods, and period increment, can be selected for the second run of strphase; these should be such that the trial period from the first run which gave a significantly smaller “string-length” than any other is at the centre of the range. The process should be continued until it is found that the period cannot be improved given the data available; at this point it will be noticed that no trial period gives a “string-length” significantly smaller than adjacent trial periods. Identify the trial period which gives the smallest “string-length” and those trial periods on either side which give “string-lengths” which are two standard deviations larger; the mid-point of this range is a good value for the period and the range limits define the two standard deviation error bars.

AS33-4 Copyright c 2012 University of Oxford, except where indicated. Analysis of eclipsing binary data

When strphase is run with the same value given for the maximum and minimum trial period, and any period increment greater than zero, phased observations are written into a ﬁle in the working directory whose name is of the form phased-observations.dat nnnnn where nnnnn is the ﬁle version number. Phased observations should be plotted using a suitable plotting program (DIPSO for example) and included with the material submitted for marking. In addition, an expanded (shorter phase range) plot is needed for the Primary Eclipse for reasons discussed below.

Should you elect to use DIPSO, there is a script for plotting the B light-curve which can be copied, along with other ﬁles needed for ﬁtting sine curves to the radial velocity data as described in the next section, using

$ cp ~aelg/eclipsing_binary/scripts/* . $ mv B_light_curve_plot.cmd P.CMD

where the second command renames B light curve plot.cmd to the more manageable P.CMD. Review the contents of P.CMD, making any necessary changes, and then produce a screen plot with

$ dipso >@P

and then, if the plot is acceptable, generate postscript output by editing P.CMD and running the script again from within DIPSO. Postscript will then be in a ﬁle called pgplot.ps in the current directory and the commands

$ open pgplot.ps $ lpr -o raw pgplot.ps

will preview it (replace “open” with “gv” if working remotely) and send it to the printer.

Having determined a period using strphase with Johnson B-band data, repeat the determination with Johnson V-band data; in order to ensure these are read into strphase, type

$ OBSERVATIONS=V.dat; export OBSERVATIONS

to redefine the OBSERVATIONS environment variable. Comment on any differences found between the two period determinations; if all agree within error limits then the period to adopt is the median between the narrower limits of the two 95% confidence intervals. A comparison should now be made with periods published in the literature and any significant differences commented on. Make a plot of the V-band light-curve as you did in the B-band case; if you used DIPSO, the script P.CMD will need to be edited but note that any previously created pgplot.ps in the same directory will be overwritten unless previously renamed.

5 Radial velocity curve ﬁtting

If two stars in a binary have purely circular orbits, their radial velocity curves will be sinusoidal. It is therefore of some interest to ﬁt functions of the form v = s + a sin(2πφ + p) (8)

Copyright c 2012 University of Oxford, except where indicated. AS33-5 Analysis of eclipsing binary data to the observed radial velocity curves for both stars. Here v is the observed radial velocity, s the systemic velocity, a the amplitude, φ the phase and p a phase offset. For circular orbits, p should not be signiﬁ- cantly different from some integral multiple of π or zero. Since v is a non-linear function of φ, unknown parameters (a, s and p) need to be determined iteratively by non-linear least squares. Non-linear least squares is an iterative procedure requiring a good starting approximation which may be made from plots of the radial velocity variation with phase. Phase both radial velocity curves with strphase but adopt the period deduced from the B and V light curves; ensure phasing compatible with that estab- lished for the light curves by using the same epoch. If the procedure described in the next paragraph is to be used, phased radial velocity data should be in ﬁles rv-1.phse and rv-2.phse for the Primary and Secondary respectively.

DIPSO may be used for radial velocity curve plotting and a script for doing so should have been copied into your directory with another script used earlier for light curve plotting. It is best to rename the required script and obtain a screen plot by

$ mv rv_curves_plot.cmd PDATA.CMD $ dipso >@PDATA which results in phased radial velocity observations for both stars (Secondary in white and Primary in red) appearing in the same diagram. If desired, a hardcopy may be obtained by editing the script and executing it once more, as described above in the case of light curve plotting.

Non-linear least squares fits to radial velocity observations are obtained with the program BNRYRVFT; this requires the preparation of a nine-line input file with a text editor. Lines in the input file are ordered as follows: name of the radial velocity data file for the Primary, an approximate amplitude for the Primary radial velocity variation in km/s, an approximate phase offset for the Primary radial velocity variation in radians, name of the radial velocity data file for the Secondary, an approximate amplitude for the Secondary radial velocity variation in km/s, an approximate phase offset for the Secondary radial velocity variation in radians, an approximate systemic velocity in km/s, the period in days and the epoch used for phasing light curve observations. Values to be entered in lines two, three, five, six and seven can be discerned from an inspection of phased radial velocity data.

Suppose the input ﬁle created as described in the previous paragraph is called binary.dat, then BN- RYRVFT is run by setting environment variables

$ DIRECTORY=‘pwd‘ $ INPUTFILE=binary.dat $ export DIRECTORY INPUTFILE and then issuing the command

$ bnryrvft which creates some screen output and the text files fort.80, fort.81, fort.90, fort.91 and binary.theory- rv. Partial derivatives of the radial velocity variation (v) in Equation 8 with respect to parameters (s), (a) and (p) are calculated analytically and checked by finite forward differencing; comments on this are provided in fort.80 and fort.90 for the Primary and Secondary respectively. Results for the fits are respectively given in fort.81 and fort.91 for the Primary and Secondary; from both files it is important to extract final values of the three elements of the vector BETA, their standard deviations and the

AS33-6 Copyright c 2012 University of Oxford, except where indicated. Analysis of eclipsing binary data

corresponding 95% conﬁdence limits. Vector elements BETA(1), BETA(2) and BETA(3) correspond to s, a and p respectively in Equation 8.

Theoretical radial velocity variations for both stars are written in the ﬁle binary.theory-rv and these need to be graphically compared with observation. A DIPSO script for doing so should already have been copied into your directory when the script for plotting light curves was copied. The procedure

$ mv rv_fits_plot.cmd PFIT.CMD $ dipso >@PFIT

is analogus to the one described above; in this hardcopy is needed, as it must be included with material presented for marking, and may be obtained in the same way. Examine the plots and the conﬁdence intervals in the phase offsets for both stars and comment on possible implications for the geometry of their orbits.

6 Light curve synthesis

Observed light and radial velocity curves are synthesised with the program NIGHTFALL (Wichmann, 2002). A user guide for NIGHTFALL can be found in the Laboratory and if absolutely necessary printed out using

$ lpr ~aelg/eclipsing_binary/nightfall/documentation/UserManual.ps

for example. Several example data and conﬁguration ﬁles are provided, and it is a good idea to play with these to gain some familiarity with the program;

In order to try NIGHTFALL out, it is suggested a new directory be created and used before NIGHTFALL is invoked by typing

$ mkdir junk $ cd junk

where the aim is to separate files generated by playing from those required for carrying out the experiment. By default NIGHTFALL will read configuration and data files from the current working directory. Example configuration and data files for trying NIGHTFALL are in different directories. To change the default behaviour of NIGHTFALL, modify environment variables by typing

$ NIGHTFALL_CFG_DIR=~aelg/eclipsing_binary/nightfall/cfg $ NIGHTFALL_DATA_DIR=~aelg/eclipsing_binary/nightfall/data $ export NIGHTFALL_CFG_DIR NIGHTFALL_DATA_DIR

and then NIGHTFALL is invoked by typing

$ nightfall -U

which produces a dialogue box for controlling program execution.

Use a configuration file to read in some data; to do this click on FILE and then Open config file ... and choose a configuration file to load. Values of various parameters will be seen to change as the chosen configuration file is loaded; if the configuration file is structured to read observation files then this will be done as well. Synthetic light and radial velocity curves are computed by clicking on Compute and then plotted by clicking on Plot. The resulting plot has two panels if observations have been read; the upper panel shows the comparison between the observed light (or radial velocity) curve and the equivalent synthetic curve, the lower panel plots residuals for each observation. Selecting Plot Options enables the plot obtained by clicking on Plot to be changed.

Before using NIGHTFALL to ﬁt observations of the eclipsing binary star of interest, it is essential to have approximate values for the unknown parameters. Assume that both stars have circular orbits whose planes lie in the line-of-sight. The orbital velocities of both stars are then obtained from ﬁts to the radial velocity curves; as the period is known, the radii of the orbits and the mass-ratio then follow. Masses are obtained by noting that the gravitational force between the two stars is also the centripetal force.

Note in particular the dependence on amplitudes derived from non-linear least squares ﬁts. The con- sequence is that formal least squares errors in the amplitudes propagate right through the analysis which follows; these are to be estimated where appropriate using

n 2 2 2 σ ( f ) = ∑(∂ f /∂xi) σ (xi) (9) i=1 which gives the variance in f (x1, x2,..., xn) if the xi are uncorrelated. The amplitudes of the two radial velocity curves would be expected to be uncorrelated since the radial velocity measurements are independent; the fitting procedure involving the use of s and p from the fit to the more massive star radial velocity curve, in fitting to the less massive star radial velocity curve, introduces weak correlation and so the above equation gives a lower limit for σ2( f ).

For circular orbits having planes in the line of sight, and making the additional assumption that spe- cific intensities at all wavelengths are the same everywhere on the stellar disk for both stars, use eclipse timings and determined orbital speeds to make rough estimates of stellar radii. If the orbits are circular, and their planes lie in the line of sight, only the larger (Primary) star is seen at Secondary eclipse; the colour index (B-V) at this phase gives the Primary effective temperature when used with a suitable calibration. Blackwell & Lynas-Gray (1998) give a calibration of effective temperature as a function of (B-V) that could be used. At quadrature (φ = 0.25 or φ = 0.75) both stars are seen. Since the fluxes contributed by the Primary in both the Johnson-B and Johnson-V bands are known, fluxes contributed by the Secondary may be estimated and its (B-V) colour deduced; this then gives the Secondary effective temperature.

When analysing the binary star of interest, it is suggested that NIGHTFALL be invoked from the same working directory that was used for the period determination. Data read by NIGHTFALL can either be the phased light and radial velocity curves output by strphase, the unphased observations read by strphase or some combination of both. Take a look at some example input data ﬁles by, for example, typing

$ cd ~aelg/eclipsing_binary/nightfall/data $ more ty_booB.dat $ more lz_cenR1.dat $ cd ~/as33 to see how NIGHTFALL input data ﬁles are arranged.

When unphased data are used by NIGHTFALL the period and epoch must be included as described in the ”User Guide”, chapter 7. Irrespective of whether phased or unphased differential photometric

AS33-8 Copyright c 2012 University of Oxford, except where indicated. Analysis of eclipsing binary data observations are used, the filter with which they were obtained must be included in the file as also described in the User Guide. In the case of radial velocity data, whether phased or not, the systemic radial velocity needs to be included in NIGHTFALL input files as described in the User Guide. It is also necessary to specify whether radial velocity data are those for the Primary or Secondary, remembering that the NIGHTFALL convention is that the Primary is the star that eclipses first; this means that what has been referred to as the Primary up to now becomes the Secondary when using NIGHTFALL and vice versa.

Having set up required data files, the next step is to establish a configuration file for your NIGHTFALL run. Again, it is worthwhile to look at some examples by typing

$ cd ~aelg/eclipsing_binary/nightfall/cfg $ more ty_boo.cfg $ cd ~/as33 to see how configuration files are structured. It is suggested that a copy of an example configuration file be made in the working directory and this copy be edited as required. Approximate values already derived are edited into the configuration file to be used as a starting point in the process of obtaining a simultaneous fit to light and radial velocity curves.

Parameters to change are MassRatio, PrimaryTemperature, SecondaryTemperature, AbsoluteMass, AbsoluteDistance, AbsolutePeriod and Name (desirable but hardly essential). The AbsoluteMass is the total mass of the two stars and AbsoluteDistance is the separation of their cen- tres of mass. In providing PrimaryTemperature and SecondaryTemperature values it is important to remember that the meaning of “Primary” and “Secondary” in NIGHTFALL is the opposite of the usual convention; the value supplied for PrimaryTemperature is therefore the approximate effective temperature derived for the Secondary and the value supplied for SecondaryTemperature is therefore the approximate effective temperature derived for the Primary. Names of available observation files should also be edited into the configuration file as indicated in the examples. Leave at least one space after each file name and code a “0” if it is to be used in the fitting process and a “1” otherwise.

Before running NIGHTFALL, remember to reset environment variables so that your conﬁguration and data ﬁles are found by typing

$ NIGHTFALL_CFG_DIR=~/as33 $ NIGHTFALL_DATA_DIR=~/as33 $ export NIGHTFALL_CFG_DIR NIGHTFALL_DATA_DIR

and invoke NIGHTFALL as before and select the prepared configuration file, being sure to check all parameters are set to values intended. Next compute synthetic light and radial velocity curves and plot these to ensure that observations have been read correctly. In particular, look at the radial velocity variations. If the observed and synthetic radial velocity variations are in anti-phase the mass ratio needs to be the reciprocal of the value specified. Also compare the observed and synthetic light curves; if there is a difference between the two outside the eclipses, it is likely that a small shift needs to be specified in the corresponding data file as described in the User Guide.

Note that NIGHTFALL specifies the sizes of stars using Roche lobe filling factors from which radii are inferred. Select Output and DataSheet to see which radii have been assigned on the basis of default Roche lobe filling factors; if these are very different from the approximate radii already estimated from the light curve eclipses, then it is very probable that they will need changing in order to obtain an improved fit. Another parameter that is not likely to be well-determined is the effective temperature of the cooler star and so this is also a candidate for adjustment at an early stage in the fitting process.

A final check is to ensure that the value of “Chi Square” printed at the bottom left of NIGHTFALL con- trol panel is non-infinite. If the value is infinite then modify the configuration file to exclude selected datasets from the fit. Once the file causing the problem is identified and corrected, other datasets previously excluded can be included again, It is a good idea to select File and Clear Memory before reading datasets, or a configuration file, for a second time.

Use of the Simplex or Simulated Annealing algorithms, provided within NIGHTFALL for optimisation, is not recommended as they require too much computer time. Faster progress can be made by manually adjusting parameters, based on physical insight, and then recomputing the synthetic light and radial velocity curves. For example, if an eclipse in the synthetic light curve is too narrow, the corresponding Roche lobe ﬁlling factor needs to be increased; if the widths of eclipses are the same, however, but a synthetic light curve eclipse is too deep then the effective temperature of the star being eclipsed needs to be decreased.

It is recommended that the best possible fit, in the minimum χ2 sense and as judged by comparing observed light (and radial velocity) curves with the corresponding synthetic curves, be obtained by simply varying the Roche lobe filling factors and the effective temperature of the cooler star; in these initial stages, retain the assumption that the plane of the orbit is in the line of sight which implies setting the orbital inclination to be i = 90o. The material presented for marking should include a table of parameter value changes and the χ2 obtained. Recording results as described makes it possible to return to an earlier stage of the fitting process if desired.

Only when a satisfactory ﬁt has been obtained is it time to improve it by selecting i < 90o; and even before this, it is essential to decide which other parameters need to be changed to compensate. In order to do so, it is suggested that preliminary values determined, on the assumption of a circular orbit whose plane is in the line of sight, be reviewed on the basis that the orbit is now inclined at angle of (90o − i) to the line of sight. Also note that if the inclination is < 90o the binary’s total mass needs to be adjusted accordingly.

If a few trial NIGHTFALL runs are made, it will also be seen that any tilt of the orbit has to be small or eclipses are no longer seen. The initial fit with i < 90o will be worse than that obtained with i = 90o but it can be improved by adjusting the Roche lobe filling factors and perhaps the temperature of the cooler star; it is recommended that these parameters be optimised again before further change is made to the orbital inclination. At least one plot of the fit to the deeper eclipse, annotated with parameter values giving that fit, is to be included in the material submitted for marking.

What are the features of the light and radial velocity curves which suggest the assumption of a circular orbit is a good one? Make some comment in the material submitted for marking which indicates how the observed radial velocity and light curves would change with increasing orbital eccentricity and how these would depend on orientation of the orbit with respect to the observer.

An assessment of errors in the Roche lobe filling factors, the effective temperature of the cooler star and the orbital inclination, can be obtained at the end of the fitting process. Simply note the largest perturbations which produce no discernable improvement in the fit. Use formal non-linear least squares errors discussed previously to derive errors in estimated masses of both stars, perturb both by the error estimates and see how the solution is changed. Material submitted for marking needs to include a table of final parameter values, including errors in them, as well as the corresponding DataSheet from NIGHTFALL. Plots of the comparison between observed and theoretical light and radial velocity curves, for the final fit only, are also required.

AS33-10 Copyright c 2012 University of Oxford, except where indicated. Analysis of eclipsing binary data

7 Comments on the binary analysed

The most important part of any academic study is the interpretation placed on the results obtained. It is suggested that a comparison be presented between the derived results and those found in the literature. If the light and radial velocity curves of R CMa have been analysed, the paper by Ribas et al. (2002) is a place to start as they summarise what is known about it and refer to earlier work. Write a couple of paragraphs to summarise what is known about the object analysed and attempt to set the results obtained in this context. In particular consider how conclusions may have been compromised by neglecting, for example, variability in either or both stars and the possible presence of a third and much fainter star in the system.

Give some consideration to whether the derived masses and radii are consistent with determined effective temperatures, on the assumption that both stars are Main Sequence objects. A convenient and modern discussion of masses and radii of normal stars is given by Torres et al. (2010); see their table 1 in particular. Main Sequence stars with the same effective temperatures as those derived for the components of R CMa are likely to be very different from the much more reliable masses estimated from ﬁts to the light and radial velocity curves. Try to think of an evolutionary scenario that explains the mass anomalies by considering past and future evolution of both stars.

Blackwell DE & Lynas-Gray AE, 1998 AAS 129, 505

Dworetsky MM, 1983, MNRAS 203, 917

Hilditch RW, 2001, An Introduction to Close Binary Stars, Cambridge University Press.

Radhakrishnan KR & Sarma MBK, 1982, Contr. Nizamiah & Japal-Rangapur Obs. No. 16.

Ribas I, Arenou F & Guinan EF, 2002, AJ 123, 2033

Torres G, Andersen J & Gimenez´ A, 2010 Astron. Astrophys. Rev. 18, 67

Tomkin J, 1985, ApJ 297, 250

Wichmann R, 2002, NIGHTFALL User’s Guide

03Oct2012 Copyright c 2012 University of Oxford, except where indicated. AS33- 12